Problem: Determine how many solutions exist for the system of equations. ${5x-y = 2}$ ${y = 8-4x}$
Convert both equations to slope-intercept form: ${5x-y = 2}$ $5x{-5x} - y = 2{-5x}$ $-y = 2-5x$ $y = -2+5x$ ${y = 5x-2}$ ${y = 8-4x}$ ${y = -4x+8}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x-2}$ ${y = -4x+8}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.